That the data are ‘really noisy’ unfortunately says nothing about the nature of the noise. If the noise is band-limited (determine this by doing a Fourier transform on it), the noise can likely be eliminated with a frequency-selective filter. For broadband noise (and your signal since the nature of the noise is not described), I would use the sgolayfilt (Savitzky-Golay filter) function. I usually use an order 3 polynomial, and then experiment with the window length until I get the result that I want.
Take the derivatives of the filtered signal.
dydx = gradient(y) ./ gradient(x);
d2ydx2 = gradient(dydx) ./ gradient(x);
This does not require that the signals be sampled consistently, however for all other signal processing (including the fft function), they must be. Use the resample function to resample them to consistent sampling intervals if they are not. To determine sampling frequency consistency, calculate:
tsstd = std(diff(x))
where ‘x’ is the independnet variable. For a consistently-sampled signal, this should be near zero (on the order of 1E-15 or so).
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